The goal of offline model-based optimization (MBO) is to propose new designs that maximize a reward function given only an offline dataset. However, an important desiderata is to also propose a diverse set of final candidates that capture many optimal and near-optimal design configurations. We propose Diversity in Adversarial Model-based Optimization (DynAMO) as a novel method to introduce design diversity as an explicit objective into any MBO problem. Our key insight is to formulate diversity as a distribution matching problem where the distribution of generated designs captures the inherent diversity contained within the offline dataset. Extensive experiments spanning multiple scientific domains show that DynAMO can be used with common optimization methods to significantly improve the diversity of proposed designs while still discovering high-quality candidates.